Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elirr | Unicode version |
Description: No class is a member of
itself. Exercise 6 of [TakeutiZaring] p.
22.
The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4280, we could redefine (df-iord 4121) to also require (df-frind 4087) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4285 (which under that definition would presumably not need ax-setind 4280 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4285. To encourage ordirr 4285 when possible, we mark this theorem as discouraged. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elirr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neldifsnd 3520 | . . . . . . . . 9 | |
2 | simp1 938 | . . . . . . . . . . 11 | |
3 | eleq1 2141 | . . . . . . . . . . . . . . . 16 | |
4 | eleq1 2141 | . . . . . . . . . . . . . . . 16 | |
5 | 3, 4 | imbi12d 232 | . . . . . . . . . . . . . . 15 |
6 | 5 | spcgv 2685 | . . . . . . . . . . . . . 14 |
7 | 6 | pm2.43b 51 | . . . . . . . . . . . . 13 |
8 | 7 | 3ad2ant2 960 | . . . . . . . . . . . 12 |
9 | eleq2 2142 | . . . . . . . . . . . . . 14 | |
10 | 9 | imbi1d 229 | . . . . . . . . . . . . 13 |
11 | 10 | 3ad2ant3 961 | . . . . . . . . . . . 12 |
12 | 8, 11 | mpbid 145 | . . . . . . . . . . 11 |
13 | 2, 12 | mpd 13 | . . . . . . . . . 10 |
14 | 13 | 3expia 1140 | . . . . . . . . 9 |
15 | 1, 14 | mtod 621 | . . . . . . . 8 |
16 | vex 2604 | . . . . . . . . . 10 | |
17 | eldif 2982 | . . . . . . . . . 10 | |
18 | 16, 17 | mpbiran 881 | . . . . . . . . 9 |
19 | velsn 3415 | . . . . . . . . 9 | |
20 | 18, 19 | xchbinx 639 | . . . . . . . 8 |
21 | 15, 20 | sylibr 132 | . . . . . . 7 |
22 | 21 | ex 113 | . . . . . 6 |
23 | 22 | alrimiv 1795 | . . . . 5 |
24 | df-ral 2353 | . . . . . . . 8 | |
25 | clelsb3 2183 | . . . . . . . . . 10 | |
26 | 25 | imbi2i 224 | . . . . . . . . 9 |
27 | 26 | albii 1399 | . . . . . . . 8 |
28 | 24, 27 | bitri 182 | . . . . . . 7 |
29 | 28 | imbi1i 236 | . . . . . 6 |
30 | 29 | albii 1399 | . . . . 5 |
31 | 23, 30 | sylibr 132 | . . . 4 |
32 | ax-setind 4280 | . . . 4 | |
33 | 31, 32 | syl 14 | . . 3 |
34 | eleq1 2141 | . . . 4 | |
35 | 34 | spcgv 2685 | . . 3 |
36 | 33, 35 | mpd 13 | . 2 |
37 | neldifsnd 3520 | . 2 | |
38 | 36, 37 | pm2.65i 600 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3a 919 wal 1282 wceq 1284 wcel 1433 wsb 1685 wral 2348 cvv 2601 cdif 2970 csn 3398 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-sn 3404 |
This theorem is referenced by: ordirr 4285 elirrv 4291 sucprcreg 4292 ordsoexmid 4305 onnmin 4311 ssnel 4312 ordtri2or2exmid 4314 reg3exmidlemwe 4321 nntri2 6096 nntri3 6098 nndceq 6100 nndcel 6101 phpelm 6352 fiunsnnn 6365 onunsnss 6383 snon0 6387 |
Copyright terms: Public domain | W3C validator |